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### Measure on the Boundary of a Hyperbolic Group

Let $\Gamma$ be a non-elementary Gromov's $\delta$-hyperbolic group. Let $B(1,n)$ be the set of elements at distance at most $n$ from the identity and let $\partial B(1,n)$ be the elements at distance ...

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**1**answer

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### Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains

Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc :
By $ \bar{P} $ , we ...

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**1**answer

180 views

### Rank of a group generated by side-pairing isometries of a polyhedron

Let $P$ be a compact convex polyhedron in $\mathbb{H}^3$. Let $G$ be a group generated by side-pairing isometries of $P$. Is there an algorithm to find the rank of $G$?

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707 views

### Non-residually finite matrix groups

By Malcev's theorem, every finitely generated linear group is residually finite (RF).
On the other hand, say, the group of rational numbers is linear, but is not residually finite. Thus, one has to ...

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**1**answer

349 views

### Relationship between hyperbolicity in group theory and hyperbolicity in geometry

Could somebody teach me about the relationship, if any, between hyperbolicity in groups (in Gromov sense) and hyperbolicity in 3-dimensional orbifolds? To be more specific, let Q be a 3-dimensional ...

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**1**answer

478 views

### The version of Montel's theorem used in the proof of Jenkins-Strebel differential

Hello,
I am afraid that my main question might be a bit too elementary, but still I ask :
In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an ...

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**0**answers

545 views

### Thales' Theorem for Hyperbolic Geometry [duplicate]

In Euclidean geometry Thales' Theorem says that if you view a diameter of a circle from any point on the perimeter it occupies exactly $90$ degrees in your field of view.
More generally for any ...

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758 views

### chromatic number of the hyperbolic plane

A notorious problem in combinatorics is the following:
If we color $\mathbb{R}^2$ so that no pair of points at unit distance get the same color, what is the fewest number of colors required?
This ...

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423 views

### How does duality of symmetric spaces explain the hyperbolic cosine theorem?

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $G/K$ is a symmetric space of noncompact type, $g=k+p$ the ...

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votes

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346 views

### center of fundamental group of finite volume-hyperbolic orbifold

Is center of a fundamental group of finite volume-hyperbolic n-orbifold trivial?
Is there a good reference that the proof is wriiten?

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162 views

### Measuring the complexity of a knot by minimum number of simplices to tile the complement

This is essentially a duplicate of: Lower bound on number of tetrahedra needed to triangulate a knot complement
Suppose a knot $K\subseteq\mathbb S^3$ is such that the complement $\mathbb ...

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**1**answer

196 views

### Mapping torus relative to an infinite orbit can be hyperbolic with finite volume?

Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the ...

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**1**answer

591 views

### Why is there a unique hyperbolic simplex of largest area?

Why is there a unqiue ideal $n$-simplex in $\mathbb H^n$ with largest volume for $n\geq 3$?
For $n=3$, this is a standard calculation, and for larger dimensions is much harder (see Haagerup and ...

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**1**answer

473 views

### Is there a simplicial volume definition of Chern Simons invariants?

Suppose we have some compact hyperbolic 3-manifold $M=\Gamma\backslash\mathbb H^3$. Now we know that the hyperbolic volume of $M$ can be defined as (a constant times) the simplicial volume of the ...

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366 views

### volume of complex hyperbolic manifolds

I would like to know if there are in the literature explicit computations of the volume of complex hyperbolic manifolds.
More precisely, let $\mathcal O$ be an imaginary quadratic number field, and ...

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**1**answer

169 views

### Example of hyperbolic 3-fold with no embedded incompressible subsurfaces

Kahn-Markovic show that every hyperbolic 3-fold contains
an immersed $\pi_1$ injective surface. Are there any known examples
of hyperbolic 3-folds that do not contain a embedded $\pi_1$ injective
...

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311 views

### Hurwitz's automorphisms theorem for infinite genus Riemann surfaces

Hurwitz's automorphisms theorem states that for a compact Riemann surface $X$ the cardinality of $Aut(X)$, the group of holomorphic automorphisms, is bounded above by $84(g(X)-1)$ and is therefore ...

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664 views

### Primitive elements in a free group of rank three

It is well-known that the fundamental group of a twice-punctured torus is a free group of rank three.
I see that there is no one-to-one correspondence between the homotopy classes of essential ...

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922 views

### Poincare Metric on Hyperbolic Plane

as is well known, we can put a metric on the upper half plane $\mathbb{R}^+ \times \mathbb{R}$
by setting
$$
d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2},
$$
...

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votes

**2**answers

650 views

### How to see isometries of figure 8 knot complement

The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...

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837 views

### How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

Let me first state the definitions :
A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ...

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501 views

### Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? [closed]

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, i.e.,
$$
S_{\Delta}=\frac{1}{2}a.h,
$$
where $a$ is the length of base and the $h$ is ...

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302 views

### Arithmetic Fuchsian group

Dear all,
I have the following questions: Are all Fuchsian groups of signature $(0;2,2,2,\infty)$ arithmetic? What is known about the trace fields of these groups?
Best, K.

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### In hyperbolic 3-orbifold with totally geodesic boundary case, is it true: rank(the fundamental group of boundary M)< or equal 2 rank(fundmental group of M)?

For a orientable three manifold M with totally geodesic boundary, this inequality is true. Because the rank of (fundemantal group of boundary M)=rank (homology group of boundary M )
then we use the ...

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**1**answer

103 views

### Will the rank of fundemantal 3 manifold be decreased is I module the n(n>1) times of a element?

I am doing some rank control about the fundamental group of a 3 dim hyperbolic orbifold. After cuting out the regular neighborhood of all the singularity, I get a manifold with imcompressible ...

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434 views

### The geometrical meaning of the common value in the law of sines in hyperbolic geometry

What is the geometrical meaning of the common value in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$ in hyperbolic geometry? I know the meaning of this ...

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869 views

### Questions on Thurston's earthquake flow

Here are some questions about the earthquake deformation of hyperbolic surface that I can't answer or find references.
I briefly recall the settings. Let's fix a closed surface $S$ with genus $g\geq ...

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208 views

### Does every hyperbolic 3 manifold with totally geodesic boundary has some finite covering space with more than one boundary component?

I am thinking about the question that: if we double a hyperbolic 3 manifold along its boundary, will the rank of fundemental group of the resulting closed manifold be strictly larger than before?\
The ...

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261 views

### Can most 3 dimensional hyperbolic orbifolds with finite volume be covered by a hyperbolic manifold?

If G is a discrete cofinite volume subgroup of PSL(2,C),then G acts on H3, H3/G is a 3-dim hyperbolic orbifold N with finite volume, my question is : Is it right in most situations that we can find a ...

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288 views

### Optimal pants decompositions of a hyperbolic surface

Let $S$ be a hyperbolic surface, which is not the punctured torus or $4$-holed sphere. I am interested in finding a ``geometrically optimal'' pants decomposition on $S$.
Here is a candidate ...

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**1**answer

427 views

### The smallest positive eigenvalue and the length of the shortest geodesic

I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two.
Let $X$ ...

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**1**answer

208 views

### Homotopy Groups of de Sitter and Anti-de Sitter?

Given $n$-dimensional de Sitter or Anti-de Sitter space, $dS_n$ or $AdS_n$, what are the homotopy groups $\pi_m(dS_n)$ and $\pi_m(AdS_n)$ and how does one calculate such things?

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454 views

### Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm

In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f ...

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219 views

### A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

Hello,
This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states :
If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...

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973 views

### Hyperbolicity on Riemann Surfaces

For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a ...

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637 views

### A question on part of “An introduction to teichmuller spaces” by Imayoshi-Taniguchi

I was reading the part on Fenchel-Nielsen coordinates, the proof on page 64, I don't understand when they say:
Since
$\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$, all $g_k$ can be glued ...

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266 views

### What is the complex structure on the boundary torus of a hyperbolic knot complement?

Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary ...

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358 views

### Is there a general dilogarithm formula for the Cheeger--Chern--Simons class?

I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm.
...

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781 views

### Hyperbolic Coxeter polytopes and Del-Pezzo surfaces

Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf
I would like to find a reference for a beautiful ...

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451 views

### Laminations as a limit of ideal triangulations

I am wondering about the following:
Suppose that $S$ is a non-compact
hyperbolic surface of finite area.
Suppose that $\lambda \subset S$ is a
non-trivial, geodesic,
measured lamination. ...

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**1**answer

356 views

### Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps )

I was studying Theorem 4.4.1 from John H. Hubbard's Teichmuller Theory, vol I, Theorem 4.4.1 ( P. 129 ) which states :
Let $X,Y$ be two hyperbolic Riemann surfaces with hyperbolic metrics $d_X,d_Y$ ...

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394 views

### a little question about Heegaard splitting

for any compact orientable hyperbolic 3-manifold with totally geodesic boundary, is there a strongly irreducible heegaard splitting?

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### Generator of translation for the hyperbolic plane? [closed]

What is the generator of translation in the Beltrami-Klein model of the hyperbolic plane?

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### How people think about ending lamination?

There are lots of works in hyperbolic 3-manifolds related to ending lamination.
But I just don't know how people think about it . what is the philosophy behind it?
maybe i should ask how people ...

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596 views

### A quick and elementary question from Hubbard's Teichmuller Theory : Volume I

Hi,
On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves :
Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U ...

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659 views

### F→E→B bundle with B,E,F hyperbolic: possible?

It would be interesting to me obtain an answer to the following easy to state question:
Does there exist a (smooth) fibre bundle $\pi\colon E\rightarrow B$ with typical fibre $F$ such that $E$, $B$ ...

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590 views

### A construction of generators of discrete subgroups of SL(2,R)

I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and ...

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641 views

### Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$

Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ ...

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### Boundary defining functions for hyperbolic surfaces

Let $M$ be a geometrically finite hyperbolic surface with one cuspidal end and one funnel end so that it can be divided into $C \cup K \cup F$ where $C$ is the cusp, $F$ the funnel and $K$ the ...

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### Definition of k -quasisymmetric maps on S^1

I know the definition of k -quasi-symmetric maps $f$ on $R$,it is
there exists $k>0$ such that $\frac{1}{k}\leq\frac{f(x+t)-f(x)}{f(x)-f(x-t)} \leq k \forall x,t\in R.$
So I just want to ...